PlanetMath: singular value decomposition

(more info)

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singular value decomposition

(Definition)

Any real $m \times n$ matrix can be decomposed into

$\displaystyle A = USV^T$

where is an $m \times m$ orthogonal matrix, is an $n \times n$ orthogonal matrix, and is a unique $m \times n$ diagonal matrix with real, non-negative elements $\sigma_i$ , $i = 1, \ldots , \min(m,n)$ , in descending order:

$\displaystyle \sigma_1 \ge \sigma_2 \ge \dots \ge \sigma_{\min(m,n)} \ge 0$

The $\sigma_i$ are the singular values of and the first $\min(m,n)$ columns of and are the left and right (respectively) singular vectors of . has the form:

$\displaystyle \begin{bmatrix}\Sigma \\ 0\end{bmatrix} \operatorname{if} m \ge n... ...torname{and}\; \begin{bmatrix}\Sigma & 0 \end{bmatrix} \operatorname{if} m < n,$

where $\Sigma$ is a diagonal matrix with the diagonal elements $\sigma_1,\sigma_2,\ldots , \sigma_{\min(m,n)}$ . We assume now $m \ge n$ . If $r=\operatorname{rank}(A) < n$ , then

$\displaystyle \sigma_1 \ge \sigma_2 \ge \cdots \ge \sigma_r > \sigma_{r+1} = \cdots = \sigma_n = 0.$

If $\sigma_r \ne 0$ and $\sigma_{r+1} = \cdots = \sigma_n = 0$ , then is the rank of . In this case, becomes an $r \times r$ matrix, and and shrink accordingly. SVD can thus be used for rank determination.